Dynamic shear amplification of reinforced concrete coupled walls
Introduction
One of the most used tools for seismic design of buildings is the spectral modal analysis (SMA), which is a method that uses the linear properties of materials to estimate the response in terms of displacements and inertial forces that a structure is subjected for seismic design. The SMA method delivers the global response through resolving N systems of one degree of freedom, and internal loads are determined by a combination method. Broadly speaking, the advantage of the SMA is that it avoids carrying out a dynamic time-history analysis (THA), such that, it evaluates the modal response for a design spectrum, combining representative periods of the structure. On the other hand, the SMA has certain limitations, since it commonly considers that the response modification factor, required to determine the design forces, applies equally to all vibration modes, while nonlinearity is commonly concentrated in the fundamental mode. Moreover, the SMA as a linear-elastic analysis, it assumes a linear behavior of the structural elements, not allowing stiffness changes due to cracking or yielding, for example.
In the SMA, the distribution of inertial forces for the linear case is commonly close to an inverted triangular (fundamental mode) shape, such that the resulting height of inertial forces is located at hef ≈ 2/3hw (hw, height of the building). If the longitudinal reinforcement at the base of a cantilever wall has yielded (nonlinear) and the wall lateral force distribution at that instant resembles more or less an even distribution in height, as shown in Fig. 1, due to the plastification at the base (effect of higher modes), it yields that hef ≈ 1/2hw. The shear amplification () for this case can be defined as the ratio between the base shear obtained from a conventional analysis (e.g., SMA) and the base shear from a nonlinear model (e.g., nonlinear time-history analysis or NLTHA), which in the case of identical base moment is equivalent to the ratio of the height of inertial forces, and would be approximately, . Larger participation of equivalent higher modes in the nonlinear analysis can lead to a resultant lateral force at an even lower level, yielding even larger shear amplification values. The general concept of dynamic shear amplification can be understood as the change in the distribution of inertial forces given the strength limitation and decrease in stiffness at the plastic hinge location (e.g., wall base). In the case of cantilever walls the only source of material nonlinearity is the plastification of the base, such that, once yielding has occurred, the effects of amplification are imminent through the presence of higher modes.
The work by Blakeley et al. [1] for cantilever reinforced concrete walls demonstrated the relevance of the higher modes in the shear response once the wall reached yielding at the base. This study was conducted with three models of 6, 15 and 20-story buildings, for 5 acceleration records of which two were constructed artificially. Its results meant a change in the seismic-resistant design provisions of New Zealand in 1982, recognizing the dynamic amplification effects for the design of reinforced concrete walls and the European-International Concrete Committee (CEB) in its 1980 edition, and currently in ACI 318 in its 2019 version. Eq. (1) that describes the dynamic shear amplification factor (), dependent on the number of floors (n), is included in the NZS 3101 [2] and ACI 318 [3] codes.
Derecho et al. [4] developed a parametric study for 10 and 40-story buildings for 6 acceleration records using a nonlinear model of the walls under a time-history analysis. The results indicate that the dynamic shear amplification depends on the rotational ductility and the fundamental period of the structure, demonstrating that this behavior is highly influenced by the level of the nonlinear incursion. Similarly, Eibl and Kentzel [5] proposed an expression to quantify the dynamic shear amplification, where the formulation assumes that only the fundamental mode is limited by the wall plastification, maintaining linear response for the second mode, standing out the effect of nonlinear response associated to the first mode. This formulation was adopted by Eurocode 8 Part 1 [6]. Priestley [7] proposed a similar formulation, but for several modes.
Moreover, dynamic shear amplification has been observed in experimental tests as well. Panagiotou and Restrepo [8] observed a base moment over-strength, defined as the ratio of the maximum measured base moment and the design base moment, of 2.7; while base shear over-strength factor was 4.2, noticing an increase of 50% for shear. Eberhard and Sozen [9] conducted small-scale dynamic tests with dual wall systems and 9 and 10-story moment frames on a shake table to observe the effect of shear amplification in reinforced concrete wall systems. In their investigation, they noted that as the system was damaged, the distribution of inertial forces behaved differently once yielding at the base was reached, generating significant increment in shear forces in the elements.
Although the dynamic shear amplification phenomenon has been known for several years, most design formulations are based on analytical developments for cantilever walls. The little information on coupled walls is usually associated with limited case studies, for symmetrical walls and considering beams as coupling elements that usually cause large variations of axial loads on walls (e.g., [10]). However, it is common to observe coupled walls or a series of walls with different dimensions coupled through beams or slabs in Chile and other places such as Europe and Japan. For this reason, this work addresses the effect of coupling on shear amplification through a series of nonlinear dynamic analysis for non-symmetric coupled walls.
Section snippets
Formulation of amplification in coupled walls
Let us consider the case of two walls with the same geometric properties and coupled with reinforced concrete slabs (Fig. 2). If a time-history linear response analysis is performed, and only earthquake actions are considered, similar values are obtained for shear and moment distribution in each of the walls. However, the coupling elements have shear forces at both ends that increase the axial load of one wall and decrease the axial load of the other wall. On the other hand, when accounting
Analysis of results
The equivalent response modification factor, Req, represents the ratio between the linear moment and the nonlinear moment of the wall at the base (Req = ML/MNL). The factor is set at the wall base, where yielding was observed on walls. Nonlinearity is based on flexural behavior in the model, which is consistent with design codes (e.g., [2], [3], [6], [22]) that promote flexural failure over shear failure. The level of nonlinearity is represented by Req, where the larger the value, the further
Shear amplification of coupled walls
To observe the dependency of the different model variables to the dynamic shear amplification (ωv*), regression (trend) lines are constructed versus the equivalent response modification factor (Req). Fig. 17a shows the regression models for the three cases that involve variations in wall height (10, 20 and 30 floors). It can be seen that there are modest differences in the amplification factors between the three cases. The largest slope of the trend lines corresponds to the 20-story models,
Conclusions
In this work, a parametric study is carried out that covers 432 nonlinear time-history analyses (including 6 records) for two coupled walls of different length (one has a fixed length of 2 m), including variations in the amount of steel ratio of the boundary element (1%, 3% and 5%), amount of steel ratio in the coupling element (0.0%, 0.3% and 0.6%), building number of stories (10, 20 and 30) and wall length (2 m, 4 m and 6 m) to capture the dynamic shear amplification. The formulation
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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